2 Background and notation

 Let F(x) be the cumulative probability distribution of a random variable X. Then the mean value for the random variable g(X) is the expectation value  

$${\sf E}g(X)\equiv \langle g(X) \rangle\equiv \int_{-\infty}^{\infty} g(x) {\rm d}F(x) \; .$$ (1)

The PDF is $p(x)={{\rm d}F(x)/{\rm d}x}$. The distribution F(x) is not necessarily smooth, so it can happen that p(x) is nonexistent at certain points. Nevertheless, the mean ${\sf E}g(X)$is defined as long as the integral in (1) exists. Following the definition (1), the k-th order moment of X is  

$$\alpha_k \equiv {\sf E}X^k =\int_{-\infty}^{\infty} x^k {\rm d}F(x) \; .$$ (2)

Thus, the mean of X is $m \equiv \alpha_1 = {\sf E}X$ and its dispersion is  

$$\sigma^2 \equiv {\sf E}(X-{\sf E}X)^2= \int_{-\infty}^{\infty}(x-m)^2 {\rm d}F(x) \; .$$ (3)

We denote the cumulative normal distribution by  

$$P(x)\equiv\int_{-\infty}^x Z(t){\rm d}t ={1\over \sqrt{2\pi}}\int_{-\infty}^x\exp(-t^2/2){\rm d}t \; ,$$ (4)

so its PDF is the Gaussian function  

$$Z(x) = { \exp(-x^2/2) \over \sqrt{2\pi} } \; .$$ (5)

We also need to recall some definitions for sets of orthogonal polynomials. Any two polynomials P_n(x) and P_m(x) of degrees $n \neq m$are orthogonal on the real axis with respect to the weight function w(x) if  

$$\int_{-\infty}^\infty w(x)P_n(x)P_m(x) = 0 \; .$$ (6)

We follow the notation of Abramowitz & Stegun (1972) where possible, so $H\!e_n(x)$ denotes the polynomial with weight function $w(x)=\exp(-x^2/2) \propto Z(x)$.According to Rodrigues' formula,  

$$H\!e_n(x) = (-1)^n {\rm e}^{x^2/2} {{\rm d}^n \over {\rm d}x^n} {\rm e}^{-x^2/2} \; .$$ (7)

We will refer to $H\!e_n(x)$as Chebyshev-Hermite polynomials following Kendall (1952). The ones with the weight $w(x)=\exp(-x^2) \propto Z^2(x)$,  

$$H_n(x) = (-1)^n {\rm e}^{x^2} {{\rm d}^n \over {\rm d}x^n} {\rm e}^{-x^2} \; ,$$ (8)

will be called Hermite polynomials.

The expansions in Chebyshev-Hermite and Hermite polynomials are used in many applications in astrophysics. For a PDF p(x) which is nearly Gaussian, it seems natural to use the expansion  

$$p(x) \sim \sum_{n=0}^{\infty} c_n {{\rm d}^n Z(x)\over {\rm d}x^n} \; ,$$ (9)

where the coefficients c_n measure the deviations of p(x) from Z(x). From the definitions (5) and (7) it follows that  

$${{\rm d}^n Z(x)\over {\rm d}x^n} = (-1)^nH\!e_n(x)Z(x) \; ,$$ (10)

so that  

$$p(x) \sim \sum_{n=0}^{\infty}(-1)^n c_nH\!e_n(x)Z(x) \; ,$$ (11)

with  

$$c_n = {(-1)^n \over n!}\int_{-\infty}^\infty p(t)H\!e_n(t) {\rm d}t \; .$$ (12)

This is the well-known Gram-Charlier series (of type A, see e.g. Cramér 1957; Kendall 1952 and references therein to the original work). Since $H\!e_n(x)$ is a polynomial, we see that the coefficient c_n in (12) is a linear combination of the moments $\alpha_k$ of the random variable X with PDF p(x). The combination is easily found by using the explicit expression for $H\!e_n(x)$,which we derive in Appendix B:, namely  

$$H\!e_n(x) = n! \sum_{k=0}^{[n/2]}{(-1)^k x^{n-2k} \over k! (n-2k)! \; 2^k } \; .$$ (13)

The Gram-Charlier series was used in cosmological applications in the paper by Scherrer & Bertschinger (1991), but note that it is incorrectly called Edgeworth expansion there. The Gram-Charlier series is merely a kind of Fourier expansion in the set of polynomials $H\!e_n(x)$.This expansion has poor convergence properties (Cramér 1957). Very often, for realistic cases, it diverges violently. We consider such an example in the next section. The Edgeworth series, discussed in detail in Sect. 5, is a true asymptotic expansion of the PDF, which allows one to control its accuracy.